J Algebr Comb (2011) 33: 277–312
DOI 10.1007/s10801-010-0245-5
The (1 − E)-transform in combinatorial Hopf algebras
Florent Hivert · Jean-Gabriel Luque ·
Jean-Christophe Novelli · Jean-Yves Thibon
Received: 2 April 2010 / Accepted: 28 June 2010 / Published online: 20 July 2010
© Springer Science+Business Media, LLC 2010
Abstract We extend to several combinatorial Hopf algebras the endomorphism of
symmetric functions sending the first power-sum to zero and leaving the other ones
invariant. As a “transformation of alphabets”, this is the (1 − E)-transform, where E
1
. In
is the “exponential alphabet,” whose elementary symmetric functions are en = n!
the case of noncommutative symmetric functions, we recover Schocker’s idempotents
for derangement numbers (Schocker, Discrete Math. 269:239–248, 2003). From these
idempotents, we construct subalgebras of the descent algebras analogous to the peak
algebras and study their representation theory. The case of WQSym leads to similar
subalgebras of the Solomon–Tits algebras. In FQSym, the study of the transformation
boils down to a simple solution of the Tsetlin library in the uniform case.
Keywords Combinatorial Hopf algebras · Symmetric functions · Derangements
F. Hivert · J.-G. Luque
LITIS, Université de Rouen, Avenue de l’université, 76801 Saint Étienne du Rouvray, France
F. Hivert
e-mail: hivert@univ-rouen.fr
J.-G. Luque
e-mail: luque@univ-rouen.fr
J.-C. Novelli · J.-Y. Thibon ()
Institut Gaspard Monge, Université de Marne-la-Vallée, 5, Boulevard Descartes, Champs-sur-Marne,
77454 Marne-la-Vallée cedex 2, France
e-mail: jyt@univ-mlv.fr
J.-C. Novelli
e-mail: novelli@univ-mlv.fr
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1 Virtual alphabets in combinatorial Hopf algebras
The notion of virtual alphabet provides a powerful symbolic notation for dealing
with endomorphisms of combinatorial Hopf algebras, at least for the combinatorial
Hopf algebras which can be realized in terms of (noncommutative) polynomials. Examples include Sym (noncommutative symmetric functions, [12, 19]), FQSym (Free
quasi-symmetric functions [9, 10], which realize the Malvenuto–Reutenauer algebra [23]), WQSym (Word quasi-symmetric functions [18]), PQSym (Parking quasisymmetric functions [28]), and their subalgebras as long as they are stable under the
internal product. These algebras can be regarded as generalizations of the Hopf algebra of symmetric functions, for which this formalism was essentially promoted by
Lascoux [20, 21].
The algebra of symmetric functions is denoted by Sym. Apart from this detail, our
notation for symmetric functions follows Macdonald’s book [22].
1.1 Virtual alphabets for symmetric functions
It is convenient to use the generic term alphabet to designate the argument of a symmetric function. Indeed, a symmetric function f is characterized by its expression
f (x1 , x2 , . . .) in terms of monomials in an infinite sequence of independent indeterminates xi , to which various sets or multisets of algebraic expressions (numbers,
monomials, cohomology classes, vector bundles, etc.) can be substituted. The diversity of possible interpretations suggests to treat as far as possible these arguments
as formal symbols (or letters, whence the term alphabet), and the possible occurrence of multiple arguments suggests to extend the usual meaning of this term and to
understand it as a multiset of symbols.
Actually, a multiset A = {a, c, c, c, f, f } is nothing but a formal linear combination of symbols with nonnegative integer coefficients, and it is more convenient to
represent it as A = a + 3c + 2f . With this notation, the union of multisets becomes
just a sum A + B, and if we have sums, we can also have differences A − B, at least
when B is contained in A.
This leads us to the point of this paragraph. When B is not contained in A, let us
call the formal combination A − B a virtual alphabet. It is easy to define the value of a
symmetric function on a virtual alphabet. Indeed, it is well known that any symmetric
function f can be expressed in terms of the elementary symmetric functions ek . The
elementary functions ek (A) of a genuine alphabet A are the coefficients of the product
(1 + ta) =
ek (A)t k
(1)
λt (A) :=
a∈A
k
and when B is contained in A, those of the alphabet C = A − B are obtained by
division of the generating functions:
λt (A − B) = λt (A)/λt (B).
(2)
When B is not contained in A, this still defines coefficients ek (A − B), which are,
by definition, the elementary symmetric functions of the virtual alphabet A − B.
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279
This defines all symmetric functions of A − B, and one has, for example, the simple
expression pn (A − B) = pn (A) − pn (B) for the power sums.
More generally, a virtual alphabet can be defined by any sequence (en (A))n≥1 ,
interpreted as its elementary symmetric functions (or as any sequence of independent
generators of the algebra of symmetric functions, such as power sums pn or complete
homogeneous functions hn ).
For example, although the exponential function et has no zeros in the complex
plane, we can introduce a virtual alphabet E such that
en (E) =
1
,
n!
λt (E) = et .
(3)
This is a useful trick, allowing to understand a lot of formulas in combinatorics or
analysis as specializations of simple identities on symmetric functions. For example,
the exponential generating function of the derangement numbers dn is just
σt (1 − E) := λ−t (1 − E)−1 =
e−t
.
1−t
(4)
Also, for a partition λ of n, n!sλ (E) = fλ , the number of standard tableaux of shape
λ (the sλ are the Schur functions). Another virtual alphabet of interest is 1 − q, where
pn (1 − q) = 1 − q n . It plays an essential role in the theory of Hall–Littlewood functions [22].
Our expression of alphabets as formal sums or differences also allows the consideration of products. For genuine alphabets A, B,
AB = {a b | a ∈ A, b ∈ B} =
a
b
a
(5)
b
and for virtual alphabets, the symmetric functions of AB are defined by any of the
formulas like pn (AB) = pn (A)pn (B) or
hn (AB) =
sλ (A)sλ (B),
(6)
λn
the famous Cauchy identity More generally, for any function f , we
have: f (AX) =
f (X) ∗ σ1 (AX), where ∗ denotes the internal product, and σt (X) = t n hn (X) is the
generating series of homogeneous complete functions.
Hence, with this formalism, we can consider transformations of alphabets on symmetric functions, which are ring homomorphisms mapping any f (X) to f (EX),
f ((1 − E)X), f ((1 − q)X), f (X/(1 − q)), and so on. Remark that in this setting,
the virtual alphabet 1 − q is the inverse of the genuine alphabet {1, q, q 2 , . . .} ≡
1 + q + q2 + · · · .
Actually, Sym is a λ-ring [20, 21], and the notion of plethysm allows much more
complicated transformations, but in the present paper, we concentrate on the above
mentioned ones since these can be defined only in terms of the (combinatorial) Hopf
algebra structure.
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1.2 Extension to combinatorial Hopf algebras
Let X and Y be two infinite alphabets. Identifying expressions like f (X)g(Y ) with
the tensor product f ⊗ g, we can regard the transformations
Δ : f (X) −→ f (X + Y )
(7)
δ : f (X) −→ f (XY )
(8)
and
as linear maps Sym → Sym ⊗ Sym, that is, as comultiplications. Moreover, both are
(obviously) algebra morphisms, and the first one being graded in the sense that
Δ : Symn −→
Symi ⊗ Symj
(9)
i+j =n
defines actually a Hopf algebra structure, whose antipode is simply f (X) → f (−X).
Clearly, the transformations f (EX), f ((1 − E)X), f ((1 − q)X and so on can be
defined only in terms of these coproducts of the antipode, and of the internal product.
It turns out that most combinatorial Hopf algebras can be realized in terms of polynomials in some infinite and totally ordered alphabet, denoted by A = {an | n ≥ 1} in
the case of noncommuting letters, and by X = {xn | n ≥ 1} in the case of commuting
letters. The basic example is the pair Noncommutative Symmetric Functions—Quasisymmetric functions (Sym, QSym) of mutually dual combinatorial Hopf algebras.
Sums and differences of alphabets are defined on both sides, and it is possible to
make sense of the product XA (see [19]). Hence, our transformations are defined in
this case.
The subject of this article is the study of the (1 − E)-transform in the pair
(Sym, QSym), and its extension to other combinatorial Hopf algebras.
2 The (1 − E)-transform in Sym
2.1 Background
Our notations for the Hopf algebra of noncommutative symmetric functions are as
in [12, 19]. This Hopf algebra is denoted by Sym, or by Sym(A) if we consider the
realization in terms of an auxiliary alphabet. Bases of its homogeneous component
Symn are labelled by compositions I = (i1 , . . . , ir ) of n. The noncommutative complete and elementary functions are denoted by Sn and Λn , and the notation S I means
Si1 · · · Sir . The ribbon basis is denoted by RI . The notation I n means that I is
a composition of n. The conjugate composition is denoted by I ∼ , the mirror image
composition by I . The descent set of I is Des(I ) = {i1 , i1 + i2 , . . . , i1 + · · · + ir−1 }.
The graded dual of Sym is QSym (quasi-symmetric functions). The dual basis of
(S I ) is (MI ) (monomial), and that of (RI ) is (FI ).
The Hopf structures on Sym and QSym allows one to partially extend the
λ-ring notation of ordinary symmetric functions (see [19], and [20] for background on
J Algebr Comb (2011) 33: 277–312
281
the original commutative version). If A and X represent totally ordered sets of noncommuting and commuting variables, respectively, the noncommutative symmetric
functions of XA are defined by
σt (XA) =
t n Sn (XA) =
→
σtx (A) =
x∈X
n≥0
t |I | MI (X)S I (A).
(10)
I
Now, X can be a virtual alphabet, defined by an arbitrary specialization of an independent set of generators of QSym. An alternative way to express the transformation
of alphabets defined by X is [19]
F (XA) = F (A) ∗ σ1 (XA),
(11)
where ∗ is the internal product. Since σ1 (XA) is grouplike, the X-transform is a
bialgebra morphism, thanks to the splitting formula
(F1 F2 · · · Fr ) ∗ G = μr (F1 ⊗ · · · ⊗ Fr ) ∗ Δr G
(12)
where in the right-hand side, μr denotes the r-fold ordinary multiplication and ∗
stands for the operation induced on Sym⊗n by ∗.
Thanks to the commutative image homomorphism Sym → Sym, noncommutative
symmetric functions can be evaluated on any element x of a λ-ring, Sn (x) being
S n (x), the nth symmetric power. Recall that x is said of rank one (resp. binomial) if
σt (x) = (1 − tx)−1 (resp. σt (x) = (1 − t)−x ). The scalar x = 1 is the only element
having both properties. We usually consider that our auxiliary variable t is of rank
one, so that σt (A) = σ1 (tA).
The argument A of a noncommutative symmetric function can be a virtual alphabet. This means that, being algebraically independent, the Sn can be specialized to
any sequence αn ∈ A of elements of any associative algebra A. Writing αn = Sn (A)
defines all the symmetric functions of A. Quasi-symmetric functions of a virtual alphabet X can be defined by a specialization of the algebraic generators of QSym,
which is more easily done by expressing the noncommutative symmetric functions
of XA in terms of A.
The specializations X = E, defined by
Sn (EA) =
(so that FI (E) =
1
n! )
and X =
Sn
1
1−q ,
A
1−q
1
S1 (A)n
n!
(13)
for which
=
1 maj(I )
q
RI (A)
(q)n
(14)
I n
are of special importance. The second one can be used to define the peak algebras and
simplifies considerably their investigation [1, 3, 24]. Here, we will study the (1 − E)transform by the methods developed in these references.
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2.2 Noncommutative derangement numbers
A possible motivation for the (1 − E)-transforms is the combinatorics of derangements. Indeed, as already mentioned, the generating function of the derangement
numbers
D(t) =
n≥0
dn
e−t
tn
=
n! 1 − t
(15)
can be expressed as
D(t) = σ1 (1 − E)t .
(16)
The specializations of the various bases of symmetric functions at 1 − E are obtained
by expanding the Cauchy kernel σ1 ((1 − E)X), and analogously, its quasi-symmetric
functions can be defined as the coefficients of the expansion of the noncommutative
Cauchy kernel
σ1 (1 − E)A = e−S1 (A) σ1 (A)
(17)
on any basis of noncommutative symmetric functions.
For example, since the quasi-monomial basis MI is dual to S I , we have
σ1 (1 − E)A =
MI (1 − E)S I (A),
(18)
S2 ((1 − E)A) = S2 (A) − S 11 (A)/2,
(19)
Sn (1 − E)A =
I
n≥0
so that
S1 (1 − E)A = 0,
S3 (1 − E)A = S3 (A) − S 12 (A) + S 111 (A)/3,
(20)
hence, implying
M1 (1 − E) = 0,
M2 (1 − E) = 1,
M11 (1 − E) = −1/2,
(21)
and
M3 (1 − E) = 1,
M12 (1 − E) = 0,
M21 (1 − E) = −1,
M111 (1 − E) = 1/3.
(22)
Since for A = 1, Sn ((1 − E)A) = dn /n!, these noncommutative symmetric functions might be called noncommutative derangement numbers (see [29] for other examples of noncommutative combinatorial numbers).
Another natural noncommutative analogue of dn is given by
λ−t (A) 1 − tS1 (A)
−1
(23)
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283
which gives back the dn for A = E, and dn (q) for A =
ribbon basis is easily seen to be
[n/2]
tn
R12i
1
1−q .
Its expansion of the
(24)
J
i=1 |J |=n−2i
n≥0
(ribbons whose first column is of even height, or permutations whose position of the
first local minimum is even; the observation that these permutations are counted by dn
is due to Désarménien [5], and an explicit bijection has been given by Désarménien
and Wachs [6, 7]).
Similarly,
λu−t (A) 1 − tS1 (A)
−1
(25)
gives a natural noncommutative analog of the generating series D(t, u) = eu−t /
(1 − t) for permutations by number of fixed points, and the expansion in FQSym
of
σt (A) 1 − t S1 (A)
−1
(26)
yields a class of permutations in bijection with arrangements: for A = E, this series
et
is 1−t
.
Denote for short by a the (1 − E) transform, i.e.,
F := F (1 − E)A = F ∗ σ1
(27)
σ1 = σ1 (1 − E)A = e−S1 σ1 ,
(28)
where
that is,
Sn =
n
Si
(−1)i 1 Sn−i .
i!
(29)
i=0
Lemma 2.1 The -transform is a projector, i.e.,
σ1 ∗ σ1 = σ1 ,
(30)
or equivalently,
Sn ∗ Sn = Sn
for all n.
(31)
Proof Since the transform is a homomorphism,
σ1 ∗ σ1 = e−S1 σ1 ∗ σ1 = e−S1 ∗ σ1 σ1 ∗ σ1
= e−S1 σ1 = σ1 .
(32)
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Hence, Sn corresponds to an idempotent δn of the descent algebra Σn . We shall
see later that the dimension of the representation CSn δn is the derangement number
dn and that δn coincides with Schocker’s derangement idempotent [32], whence the
name derangement algebra below for the corresponding ideal of the descent algebra.
2.3 The small derangement algebra D(0)
2.3.1 Definition of D(0)
Imitating the construction of the peak algebra in [3], we define the small derangement
algebra, or the derangement ideal as
D(0) := Sym = Sym(A) ∗ σ1 .
(33)
By definition of the transformation of alphabet, D(0) is a Hopf subalgebra of Sym,
(0)
and thanks to the second equality, each homogenous component Dn is a left ideal
(0)
of Symn for the internal product, explaining the two names for D , depending on
which property we want to emphasize.
(0)
2.3.2 Dimension and bases of Dn
We have S1 = 0, and the other Sn are clearly algebraically independent. Hence, the
(0)
(0)
dimension dn of Dn is given by the number of compositions of n with no part
equal to 1. These elements satisfy the induction
(0)
d0 = 1,
(0)
d1 = 0,
(0)
(0)
dn(0) = dn−1 + dn−2
for n ≥ 2,
(34)
hence are shifted Fibonacci numbers.
Let us say that a composition is nonunitary if it does not contain the part 1. Then
(0)
the S I with I nonunitary form a basis of Dn .
Since the coproducts
Si ⊗ Sj
(35)
ΔSn =
i+j =n
are given by the same formula as those of the Sn , we see that Dn(0) is isomorphic to
the quotient Sym of Sym by the two-sided ideal generated by S1 . Hence, its (graded)
dual is isomorphic to the subalgebra QSym of QSym spanned by the MI for I
nonunitary.
We shall see below (Proposition 2.2) that σ1 is neutran in Dn(0) . Since σ1 is a
projector, we have
f ∗ g = (f ∗ g) .
(0)
(36)
Sym being stable under the internal product, Dn is also ∗-isomorphic to Symn .
Recall that the ribbon basis of Sym is given by
RI =
SJ
(37)
J ≤I
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285
where ≤ is the reverse refinement order. If I is nonunitary, so are all J ≤ I , thus
QI := RI
(1 ∈
/ I)
(38)
(0)
is a basis of Dn .
(0)
2.4 Algebraic structure of (Dn , ∗)
Since the construction of D(0) mimics the one of the peak ideal, obtained in [3] as
the image of the (1 − q)-transform at q = −1, one may expect that D(0) shares many
properties with the peak ideal. There are some differences, however. Whilst the peak
ideal has no unity for the internal product ∗, we have the following proposition.
(0)
Proposition 2.2 For all n, (Dn , ∗) is a unital algebra with Sn as neutral element.
Proof From Lemma 2.1, we already know that σ1 is neutral on the right. To prove
that it is neutral on the left, let us consider its action on the generating series of a basis
of D(0)
σ1 X · (1 − E)A =
MI (X)S I (A).
(39)
I
We have
σ1 ∗ σ1 X · (1 − E)A = σ1 ∗ σ1 (XA) ∗ σ1
(40)
and
σ1 ∗ σ1 (XA) = e−S1 (A) σ1 (A) ∗ σ1 (XA)
= μ e−S1 (A) ⊗ σ1 (A) ∗2 σ1 (XA) ⊗ σ1 (XA)
= e−S1 (A) ∗ σ1 (XA) σ1 (A) ∗ σ1 (XA)
= e−S1 (XA) σ1 (XA)
(41)
so that
σ1 ∗ σ1 X · (1 − E)A = e−S1 (XA) σ1 (XA) ∗ σ1
= e−S1 (X·(1−E)A) σ1 X · (1 − E)A
= σ1 X · (1 − E)A
since S1 (X · (1 − E)A) = 0.
(42)
(0)
2.5 Representation theory of Dn
(0)
Now that we know that Dn is unital, we can investigate its representation theory.
We first look at the idempotents.
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(0)
2.5.1 Idempotents in Dn
Recall that the (right) Zassenhaus idempotents ζn are defined as the homogeneous
elements of degree n (that is, ζn ∈ Symn ) satisfying (see [19]):
σ1 =:
e ζk = e ζ1 e ζ2 e ζ3 . . . .
(43)
k≥1
Obviously, ζ1 = S1 so that ζ1 = 0. Since the -transform is multiplicative,
σ 1 = e ζ2 e ζ3 . . .
(44)
σ1 = e−ζ1 σ1 = eζ2 eζ3 . . .
(45)
but also
so that
ζi
= ζi . Extracting the term of degree n, we have
Sn =
ζλ
,
mλ
(46)
|λ|=n
1∈λ
/
where for a partition λ = (λ1 ≥ λ2 ≥ · · · ), ζ λ := ζλ1 · · · ζλr , and mλ := i≥1 mi (λ)!,
where mi (λ) the multiplicity of i in λ.
ζλ
We shall make use of the notation eλ := m
. For a composition I , we set ζ I =
λ
ζi1 . . . ζir and mI = mλ if I ↓= λ.
2.5.2 A basis of idempotents
The ζn are primitive elements with commutative image pn /n, hence are Lie idempotents [19]. As with any sequence of Lie idempotents, we can construct an idempotent
basis of Symn from the ζn .
We need the following lemma from [24], easily derived from the splitting formula
(compare [19, Lemma 3.10]).
Recall that the radical of (Symn , ∗) is Rn = R ∩ Symn , where R is the kernel of
the commutative image Sym → Sym.
Lemma 2.3 Denote by S(J ) the set of distinct rearrangements of a composition J .
Let I = (i1 , . . . , ir ) and J = (j1 , . . . , js ) be two compositions of n. Then
(i) if (J ) < (I ) then ζ I ∗ ζ J = 0.
(ii) if (J ) > (I ) then ζ I ∗ ζ J ∈ Vectζ K : K ∈ S(J ) ∩ R. More precisely,
ζI ∗ ζJ =
J, J1 · · · Jr ΓJ1 · · · ΓJr
(47)
J1 ,...Jr
|Jk |=ik
where for a composition K of k, ΓK := ζk ∗ ζ K , and
denotes the shuffle
products of compositions regarded as words over the positive integers.
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287
(iii) if (J ) = (I ), then ζ I ∗ ζ J = 0 only for J ∈ S(I ), in which case ζ I ∗ ζ J =
mI ζ I .
Corollary 2.4
(i) The elements
eI =
1 I
ζ ,
mI
I n,
(48)
are all idempotents and form a basis of Symn . This basis contains in particular
a complete set of minimal orthogonal idempotents, eλ of Symn .
(0)
(ii) The eI such that I does not have a part equal to 1 form a basis of Dn .
(iii) The eλ with no part equal to 1 in λ form a complete set of minimal orthogonal
idempotents of Dn(0) .
(0)
2.6 Cartan invariants of Dn
(0)
By (iii) of Lemma 2.3, the indecomposable projective module Pλ = Dn ∗eλ contains
/ S(λ), (i) and (ii) imply that eI ∗ eλ is in Vectζ K :
the eI for I ∈ S(λ). For I ∈
K ∈ S(λ). Hence, this space coincides with Pλ . So, we get immediately an explicit
decomposition
Dn(0) =
Pλ ,
Pλ =
CeI .
(49)
λn, 1∈λ
/
I ∈S(λ)
The Cartan invariants
cλ,μ = dim eμ ∗ Dn(0) ∗ eλ
(50)
are now easily obtained. The above space is spanned by the
eμ ∗ eI ∗ eλ = eμ ∗ eI ,
I ∈ S(λ).
(51)
From (ii) of Lemma 2.3, this space has the dimension of the space [S μ (L)]λ , spanned
by all symmetrized products of Lie polynomials of degrees μ1 , μ2 , . . . formed from
ζi1 , ζi2 , . . . , as in the classical result of Garsia–Reutenauer for the descent algebra [15].
In the following examples, partitions are ordered by reverse lexicographic order.
For n ≤ 4, the Cartan matrix of Dn0 is trivial: it is the identity matrix of size dn(0) , since
there is at most one value in a partition of n with no part one. For n up to 7, the Cartan
invariants are given by the matrices (60) and (61) at q = 1. Indeed, the q-analogues
defined from the Loewy series can be explicitly calculated.
2.7 Quiver and q-Cartan invariants (Loewy series)
We shall use the following modified refinement order on partitions: we write λ ≺p μ,
and say that λ is p-finer than μ, if each part of μ is either a part of λ or a sum of
distinct parts of λ. Hence, μ covers λ iff μ is obtained from λ by merging two distinct
parts.
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Still relying upon point (ii) of the lemma, we see that cλ,μ = 0 if λ is not p-finer
than (or equal to) μ, and that if μ is obtained from λ by adding up two parts λi , λj ,
eμ ∗ eI = 0 if λi = λj and is a nonzero element of the radical otherwise.
The q-analogues of the Cartan invariants
(52)
q k dim eμ ∗ radk Dn(0) ∗ eλ / radk+1 Dn(0)
cλ,μ :=
k
can now be obtained from Proposition 2.2 and the following lemma.
Lemma 2.5 Let A be an associative algebra. If e is an idempotent of A such that e
is neutral in B = Ae, then
radk B = e radk A e.
(53)
Proof Let x ∈ rad B. There exists an integer n such that (xB)n = 0. Then
(xeAe)n = (xA)n e = 0,
(54)
(xA)n exA = (xA)n xA = (xA)n+1 = 0.
(55)
so that, as well
Thus, the right ideal xA is nilpotent, which proves that x ∈ rad A. Since x = exe,
x ∈ e rad Ae and we have shown that rad B ⊆ e rad Ae.
Conversely, if x ∈ rad A, so that x n = 0 for a certain n, then (exe)n = x n e = 0,
whence exe ∈ rad B, which proves the claim for k = 1.
Now, if x ∈ radk B, x = x1 . . . xk with xi = eyi e, for some yi ∈ rad A. Hence,
x = ey1 e . . . eyk e = ey1 y2 . . . yk e ∈ e radk Ae.
(56)
Conversely, any x of the form ey1 . . . yr e with yi ∈ rad A can be rewritten as x =
ey1 e . . . eyk e ∈ radk B.
Applying this to A = Symn , and e = Sn , we obtain from the known description of
the q-Cartan matrices of Symn [31]:
Theorem 2.6
(0)
(i) In the quiver of Dn , there is an arrow λ → μ iff μ is obtained from λ by adding
two distinct parts.
(0)
(ii) The q-Cartan invariants of Dn are given by
cλ,μ (q) = cλ,μ q
(λ)− (μ)
.
(57)
if λ is finer than or equal to μ, and cλ,μ (q) = 0 otherwise.
The result can also be derived as follows: In [4], it is shown that the powers of the
radical of Symn for the internal product coincide with the homogeneous component
J Algebr Comb (2011) 33: 277–312
289
of degree n of the lower central series of Sym for the external product:
R∗j = γ j (Sym)
(58)
where γ j (Sym) is the ideal of Sym generated by the commutators [Sym,
γ j −1 (Sym)].
Since D(0) is a free associative algebra over a sequence of primitive elements
(ζk )k≥2 with the same internal product as in Sym, the argument of [4] can be reproduced verbatim, and we see that
rad D(0)
∗j
= γ j D(0) = R∗j ∩ D(0) .
(59)
This shows that, in D(0) as well as in Sym, for λ finer than μ, eμ ∗ eI is nonzero
modulo rad∗2 iff μ is obtained from λ by summing two distinct parts. And more
generally, eμ ∗ eI is in rad∗k and nonzero modulo rad∗k+1 iff (λ) − (μ) = k.
(0)
2.8 Tables of the q-Cartan invariants of Dn
The labels for row and columns of the q-Cartan matrices, namely partitions with no
part one, are in reverse lexicographic order. In the following matrices, the zero entries
are represented by dots to enhance readability.
(0)
(0)
For n ≤ 4, the q-Cartan matrix of Dn is the identity matrix of size dn . The first
(0)
(0)
(0)
non-trivial example arises for n = 5. The q-Cartan matrices of D5 , D6 , and D7
are respectively
⎛
1 q
⎜. 1
⎜
⎝. .
. .
1 q
. 1
(0)
.
.
1
.
⎞
.
.⎟
⎟
.⎠
1
⎛
1 q
⎜. 1
⎜
⎝. .
. .
q
.
1
.
⎞
q2
q⎟
⎟
.⎠
1
(60)
(0)
and those of D8 and D9 are
⎛
1 q
⎜. 1
⎜
⎜. .
⎜
⎜. .
⎜
⎜. .
⎜
⎝. .
. .
q
.
1
.
.
.
.
. q2
. q
. .
1 .
. 1
. .
. .
q2
.
q
.
.
1
.
⎞
.
.⎟
⎟
.⎟
⎟
.⎟
⎟
.⎟
⎟
.⎠
1
⎛
1 q
⎜. 1
⎜
⎜. .
⎜
⎜. .
⎜
⎜. .
⎜
⎜. .
⎜
⎝. .
. .
q
.
1
.
.
.
.
.
q
.
.
1
.
.
.
.
(0)
q2
q
.
.
1
.
.
.
2q 2
q
q
q
.
1
.
.
⎞
. q3
. q 2⎟
⎟
. .⎟
⎟
. .⎟
⎟.
. q⎟
⎟
. .⎟
⎟
1 .⎠
. 1
(61)
On these matrices, one can read the quiver of Dn . Note that it is a subquiver of
the quiver of Symn (see [31]), since one cannot create parts 1 by merging parts of
non-unitary partitions.
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2.9 The (large) derangement algebra D = D(∞)
Pursuing the analogy with the peak algebra, let us define
D :=
Sk D(0) .
(62)
k≥0
Note already that D is not a subalgebra of Sym. It is only a sub-coalgebra. Moreover,
we have the following theorem.
Theorem 2.7 Each homogeneous component Dn of D is stable under ∗. It is a unital
algebra, since it contains Sn , the neutral element of ∗ in Sym.
We shall prove a slightly more general result, interpolating between D(0) and D.
2.10 A filtration of D
(k)
Define Dn by
Dn(k)
:=
k
(0)
Sj Dn−j .
(63)
j =0
(0)
For k = 0, this is Dn and for k ≥ n, one recovers Dn .
The following alternative definition of the filtration will be useful in the sequel:
Lemma 2.8
Dn(k) =
min(n,k)
j
(0)
S1 Dn−j .
(64)
j =0
Proof Expanding σ1 = eS1 σ1 , we see that
Sk
Sk ≡ 1
k!
mod
(0)
Sj Dn−j .
(65)
j <k
(k)
2.11 Dimensions of the Dn
(k)
(k)
From (63), we see that the dimension dn of Dn is
(0)
dn(k) =
dn−i .
i≤k
For k = ∞, these are the usual Fibonacci numbers.
The first values are given in Table 1.
(66)
J Algebr Comb (2011) 33: 277–312
Table 1
291
0
1
2
3
4
5
dn
(0)
1
0
1
1
2
3
5
(1)
dn
(2)
dn
(3)
dn
(∞)
dn
1
1
1
2
3
5
1
1
2
2
4
1
1
2
3
1
1
2
3
n
6
7
8
9
8
13
21
8
13
21
34
6
10
16
26
42
4
7
11
18
29
47
5
8
13
21
34
55
2.12 The complete picture
(k)
Theorem 2.9 For all n and k, each homogeneous component Dn of D(k) is stable
under ∗. It is a unital algebra and the neutral element is
Pn(k) :=
min(k,n)
i=0
S1i S .
i! n−i
(67)
The theorem is a consequence of the following lemma.
Lemma 2.10 Let f and g be in Sym. Then
m
n
0
S1 S1 f ∗
g = Sn 1
m!
n!
n! (f ∗ g )
if m = 0,
(68)
otherwise.
Proof Replacing the right factor by its generating series, we have
S1m ∗ eS1 g(1) f ∗ eS1 g(2)
S1m f ∗ eS1 g =
(g)
=
eS1 g(1) ∗ S1m f ∗ eS1 g(2) ,
(69)
(g)
since S1m is central for ∗. Now, (eS1 S I ∗ S1m ) = 0 if I is not empty since Sn ∗ S1n =
S1n ∗ Sn = 0 for n ≥ 1 as e−S1 ∗ σ1 = 1. So, the whole sum reduces to
S1m f ∗ eS1 g .
(70)
Thus,
f ∗ eS1 g = f ∗ σ1 ∗ eS1 g .
Consider now the generic case g = σ1 (XA):
σ1 ∗ eS1 σ1 (XA) = e−S1 σ1 ∗ eS1 σ1 (XA)
= e−S1 ∗ eS1 σ1 (XA) eS1 σ1 (XA)
(71)
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J Algebr Comb (2011) 33: 277–312
= eS1 ∗ e−S1 σ1 (XA) ∗ e−S1 eS1 σ1 (XA)
= e−S1 e−S1 ∗ σ1 (XA) eS1 σ1 (XA)
= e−S1 e−S1 (XA) eS1 σ1 (XA)
= e−S1 eS1 σ1 (XA) = σ1 (XA)
(72)
where the third and fourth equalities come from the fact that e−S1 is central for ∗ and
idempotent. Multiplying by f on the left yields
f ∗ eS1 g = f ∗ g ,
(73)
whence the statement.
(k)
(k)
Proof of the theorem From Lemma 2.8, we have Pn ∈ Dn . Moreover, since the
(k)
S1k S I with I nonunitary form a basis of Dn adapted to the direct sum decomposi
(k)
tion, that Pn is neutral on both sides is equivalent to the already known fact that Sn
(0)
is neutral in Dn .
(0)
(m)
Corollary 2.11 The map φm : Dk → Dk+m defined by
φm (f ) =
S1m
f
m!
(74)
is a (nonunital) monomorphism of algebras.
Corollary 2.12 As an algebra, Dn is isomorphic to the direct sum
Dn ∼
n
(0)
Dn−k .
(75)
k=0
(k)
Proof As a vector space, Dn is the direct sum of the spaces Vn
(k)
( )
(0)
= S1k Dn−k . By
(k)
Lemma 2.10, Vn ∗ Vn = 0 for k = , and by Corollary 2.11, each Vn
(0)
gebra isomorphic to Dn−k .
is a subal
(k)
2.13 Representation theory of Dn
(k)
We are now in a position to deduce the representation theory of the Dn from that
(0)
of Dn . We can extend Corollary 2.4 to all values of k, as a direct consequence of
Lemma 2.3.
Corollary 2.13 Let eI be the idempotents of Sym defined in Corollary 2.4.
(i) The eJ such that J = (1j , I ) with j ≤ k and I does not contain a part 1 form a
(k)
basis of Dn .
(k)
(ii) The principal idempotents of Dn are the eλ such that m1 (λ) ≤ k.
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293
(k)
We can now state the general result on the representation theory of Dn :
(k)
Corollary 2.14 The irreducible representations of Dn are of dimension 1 and parameterized by partitions of n with at most k parts equal to 1.
Note that the principal idempotents of Dn are also a complete set of minimal
orthogonal idempotents of Symn .
2.14 q-Cartan matrix of Dn(k)
We order the labels of the rows and columns of the q-Cartan matrices, namely partitions with at most k ones, first by their number of ones and then in reverse lexicographic order. So, for example, with n = 5 and k = 3, the order is
[5, 32, 41, 221, 311, 2111].
(76)
(k)
With this convention, the q-Cartan matrix of Dn is the block-diagonal matrix ob(0)
tained by putting on the diagonal the q-Cartan matrices of Dn−i for i ≤ min(k, n).
Indeed, from the previous results, one easily sees that
Lemma 2.15
Dn(k) = Pn(k) ∗ Symn ∗ Pn(k) .
(k)
Since each Pn
(77)
is a sum of orthogonal idempotents (the S1i Sn−i /i!), this proves
(∞)
that the q-Cartan matrix of Dn is deduced from the q-Cartan matrix of Symn by
putting to 0 the entries whose row and column do not have the same number of ones.
2.15 Block projectors
j
(0)
(k)
We have seen that each space S1 Dn−j has as unit a part of Pn , namely
Dn,k =
S1k S .
k! n−k
(78)
Their double generating series is
D(t, u; A) :=
n
t n−k uk Dn,k (A) = e(u−t)S1 (A) σt (A),
(79)
n≥0 k=0
that is, the noncommutative analog of the generating function of dn,k , the number of
permutations in Sn with exactly k fixed points:
D(t, u) =
n
1 eu−t
.
dn,k uk t n−k =
n!
1−t
n≥0
k=0
(80)
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2.16 q-dimension polynomials
(∞)
If one sums up the entries of the q-Cartan matrix of Dn
polynomials in q, refining the Fibonacci numbers:
, one gets the following
1, 2, 3, 5, q + 7, 2 q + 11, q 2 + 5 q + 15, 3 q 2 + 9 q + 22, . . .
(81)
better represented in the following triangle:
1
2
3
5
7
11
15
22
30
42
56
77
101
135
176
231
1
2
5
9
17
28
47
73
114
170
253
365
1
3
7
1
16
3
31
9
1
58 21
4
102 47
12
1
175 94
32
4
286 183 74 14 1
461 333 162 40 5
(82)
The first column (constant terms of the polynomials) corresponds to the size of the
matrix, hence the number of partitions of n. The second column is sequence A000097
of [35]: from the characterization of the quiver of Symn , we also have that the second
column gives to the number of ways of selecting two different parts different from 1,
in all partitions of n. Finally, it is also equal to the number of ways of selecting two
different parts, in all partitions of n − 2, hence justifying that the number of arrows
(∞)
in the quiver of Dn is equal to the number of arrows in the quiver of Symn−2 .
3 The (1 − E) transform in WQSym∗
3.1 Word quasi-symmetric functions
A word u over N∗ is said to be packed if the set of letters occurring in u is an interval
of N∗ containing 1. The algebra WQSym(A) (Word Quasi-Symmetric functions)
is defined as the subalgebra of KA based on packed words and spanned by the
elements
w,
(83)
Mu (A) :=
pack(w)=u
J Algebr Comb (2011) 33: 277–312
295
where pack(w) is the packed word of w, that is, the word obtained by replacing all
occurrences of the k-th smallest letter of w by k. For example,
pack(871883319) = 431442215.
(84)
This is the invariant algebra of the quasi-symmetrizing action of S(A) on KA
[10]. Packed words can be identified with set compositions in an obvious way, and
geometrically, they can be interpreted as facets of the permutohedron: a packed
word w = w1 . . . wn with largest entry can be identified with the set composition
[P1 , . . . , P ] where Pj = {i ≤ n | wi = j }. For example, 431442215 corresponds to
[{3, 8}, {6, 7}, {2}, {1, 4, 5}, {9}].
Let Nu = M∗u be the dual basis of (Mu ). It is known that WQSym is a self-dual
Hopf algebra [18, 27] and that on the graded dual WQSym∗ , an internal product ∗
may be defined by
Nu ∗ Nv = Npack(u,v) ,
(85)
where the packing of biwords is defined with respect to the lexicographic order on
biletters, so that, for example,
42412253
pack
= 62513274.
(86)
53154323
This product is induced from the internal product of parking functions [25, 26,
28] and allows one to identify the homogeneous components WQSymn with the
(opposite) Solomon–Tits algebras, in the sense of [30].
The (opposite) Solomon descent algebra, realized as Symn , is embedded in the
(opposite) Solomon–Tits algebra realized as WQSym∗n by
Nu ,
(87)
SI =
ev(u)=I
where ev(u) is the evaluation of u, i.e., the vector whose ith component is the number
of occurrences of i in u.
From now on, we shall denote WQSym∗ by W.
3.2 Idempotents of W
3.2.1 The semisimple quotient
It is known that the radical of Wn is spanned by the differences
Nu − Nv ,
(88)
where v = σ (u) for some permutation σ of the support of u, supp(u) = {i | |u|i = 0}.
This is easily seen: Equation (85) implies that the Nu − Nv are nilpotent of order
2 if v = σ (u), and that their span is an ideal Rn . Moreover, any product of n such
factors Nui − Nvi vanishes since the product of two such factors is either strictly
finer than ui or zero, so that Rn is nilpotent. The quotient Wn /Rn is semisimple. It
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J Algebr Comb (2011) 33: 277–312
can be identified with the (commutative) algebra of set partitions with ∧ (the inf for
the refinement order on set partitions) as product. Indeed, packed words encode set
compositions, u = u1 . . . un corresponding to the set composition of [n] in which i
belongs to the block ui , e.g.,
u = 21231 ⇐⇒ {2, 5}, {1, 3}, {4} ,
(89)
and the left action of permutations amounts to permuting the blocks, e.g., with
σ = 231,
σ (21231) = 32312 ⇐⇒ {4}, {2, 5}, {1, 3} .
(90)
Hence, the idempotents of a complete family of Wn are parametrized by set partitions of [n].
3.2.2 The idempotents of Saliola
In [31], Saliola has given a general recipe for constructing such complete sets. Given
a packed word u, denote by Π(u) the set partition obtained by forgetting the order
among the blocks of the set compositions encoded by u.
For each set partition π of [n], choose a linear combination
lπ =
cu N u ,
(91)
Π(u)=π
where the coefficient cu depends only on the evaluation ev(u) of u, and
cu = 1.
(92)
Π(u)=π
Start with the initial condition
e{1},{2},...,{n} =
1 Nσ ,
n!
(93)
σ ∈Sn
hence equal to S1n /n! in Sym, and define by the induction
eπ = lπ ∗ N1n −
eπ
(94)
π >π
where N1n = Sn is the identity of ∗, and ≥ is the reverse refinement order. Then the
eπ form a complete set of orthogonal idempotents of Wn .
3.2.3 A nonrecursive construction
Families of Saliola idempotents can be computed for all Wn simultaneously, in a nonrecursive way from families of idempotents of the descent algebra Symn constructed
by the method developed in [19]. Recall that the starting point of this construction is
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297
a sequence of Lie idempotents γn ∈ Symn , that is, an arbitrary sequence of primitive
elements whose commutative image in Sym is pn /n.
Then if we decompose the identity Sn of Symn as
cI γ I ,
(95)
Sn =
I n
the elements
eλ :=
cI γ I ,
(96)
I ↓λ
where I ↓ λ means that the nondecreasing rearrangement of the composition I is the
partition λ, with λ a partition of n, form a complete family of orthogonal idempotents
of Symn .
Let us fix such a family, and define for each set partition π of [n]
lπ =
cev(u) Nu .
(97)
Π(u)=π
These elements satisfy Saliola’s conditions: obviously, cev(u) depends only on ev(u),
and
cev(u) =
mi (λ)!
cI
(98)
Π(u)=π
i
I ↓λ
is the coefficient of pλ /zλ in the commutative image of Sn , which is hn , so it is 1.
Hence, the sequence (γn ) determines idempotents eπ of the Wn by the recursion (94).
But we can also compute these directly as follows.
Theorem 3.1 The idempotents eπ are given by the internal products
e π = lπ ∗ e λ .
(99)
Proof Let ẽπ = lπ ∗ eλ . For π = {{1}, . . . , {n}}, we have lπ = S1n /n!, eλ = S1n /n!, so
that ẽπ = S1n /n!.
Let lλ = Λ(π)=λ lπ , where Λ(π) is the integer partition recording the block
lengths of π . We have
lλ =
cI S I ≡ eλ mod
Kγ I ,
(100)
I ↓λ
l(I )>l(λ)
so that eλ = lλ ∗ eλ in Sym.
We want to show that ẽπ = eπ . For that, recall from [31] that, if Π(u) ≤ π , Nu ∗
eπ = 0, so
that lπ ∗ eπ = 0 if π > π , and Λ(π ) = Λ(π). This implies in particular
n
that eλ =
Λ(π)=λ eπ . Indeed, this is true for λ = 1 , and, by induction, eπ = lπ ∗
(N1n − l(π )>l(π) eπ ), since π > π implies l(π ) > l(π). Hence
eπ = lπ ∗ N1n −
e λ = lπ ∗
eλ .
(101)
l(λ )>l(λ)
l(λ )≤l(λ)
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J Algebr Comb (2011) 33: 277–312
Summing over π , we get
e π = lλ ∗
Λ(π)=λ
eλ = eλ .
(102)
l(λ )≤l(λ)
Now,
eπ = lπ ∗ N1n −
eπ
eπ
π >π
π >π
= lπ ∗ e λ +
= lπ ∗
eπ
= lπ ∗ eλ = ẽπ .
π >π;Λ(π )=Λ(π)
(103)
3.3 The (1 − E)-transform in W
The embedding (87) of Sym in W can be defined on the generators as
Sn −→ N1n .
(104)
It is clearly a bialgebra morphism. The element
σ1 = e−N1
N1n
(105)
(106)
n≥0
is well defined in W, and so is the -transform
F := F ∗ σ1 .
3.4 Bases of W Let us say that a packed word u is nonunitary (and unitary otherwise) if no letter
occurs exactly once in u. These words correspond to set compositions without singletons.
Proposition 3.2 The Nu with u nonunitary form a basis of W .
Proof Let us say that v is finer than u (and write v > u) if the set composition encoded by v is finer than the set composition encoded by u. Then
Nu ∗ σ1 = Nu +
cuv Nv ,
(107)
v
where v > u or v is unitary. Hence, the Nv with u nonunitary are linearly independent.
J Algebr Comb (2011) 33: 277–312
299
3.5 Algebraic structure of W Let J be the two-sided ideal of W generated by the Nu such that u has at least a
letter occurring exactly once. The product rule (85) shows that J is an ideal for the
internal product as well. Hence, the projection
π : W −→ W/J
(108)
is a morphism for ∗. Its restriction to W is then an isomorphism, and clearly,
π σ1 = σ1 .
(109)
Since σ1 is neutral in W/J , we have the following proposition.
Proposition 3.3 σ1 is neutral in W .
Note that this proof would apply to Sym as well. To summarize, we have the
following proposition.
Proposition 3.4 W is isomorphic to W/J as a Hopf algebra, and each Wn is
∗-isomorphic to Wn /Jn , with Dn = Sn = N1n as neutral element.
3.6 Representation theory of W We can now apply Lemma 2.5 with A = Wn , B = Wn and e = Sn .
Thus,
radk Wn = Dn ∗ radk Wn ∗ Dn .
(110)
The irreducible representations of Wn , which are one-dimensional are parameterized by set partitions of [n].
The q-Cartan matrices and quiver of Wn have been determined in [33]:
cα,β (q) = cα,β q l(α)−l(β) ,
(111)
where l(π) is the number of blocks of a set partition π , and the Cartan invariant cα,β
is 0 if α is not finer than β, and otherwise
(112)
cα,β = (mi − 1)!,
i
where for each block Bi of β, mi is the number of blocks of α into which Bi has been
split.
For example, with α = 12|3|4|56|7 and β = 1234|567, we have: cα,β =
(3 − 1)!(2 − 1)! = 2, l(α) = 5, l(β) = 2, so that cα,β (q) = 2q 3 .
Theorem 3.5 The q-Cartan matrix of Wn is the restriction to rows and columns
indexed by nonunitary set partitions of [n] of (111). In particular, the vertices of
the quiver of Wn are the non-unitary set partitions, and there is an arrow α → β
whenever β is obtained from α by merging two blocks.
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J Algebr Comb (2011) 33: 277–312
3.7 Analogue of D in W
(0)
Let Vn = Wn and
Vn(k) =
n
Dn,k ∗ Wn ∗ Dn,k .
(113)
k=0
(k)
Then, as in the case of Sym, each Vn is a unital subalgebra of Wn .
4 The (1 − E)-transform in FQSym
4.1 Definition
Recall that FQSym is based on permutations, that in the mutually dual bases Fσ =
Gσ −1 , the internal product is defined by
Fσ ∗ Fτ = Fσ τ
or equivalently Gσ ∗ Gτ = Gτ σ ,
(114)
and that Sym is embedded into FQSym by Sn = G12...n . The transformation can
therefore be defined by
Fτ := Fτ ∗ σ1 .
(115)
Since the splitting formula remains valid in FQSym when the right factor of the
internal product is in Sym [10], this is again a Hopf algebra morphism.
As we shall see below, in FQSym, the idempotent Dn = Sn as well as the other
Dn,k defined in (78) admit an interesting interpretation.
4.2 The Tsetlin library (uniform case)
The (1 − E)-transform in FQSym is related to a classical problem in probability
theory known as the Tsetlin library (see e.g., [2]). This is a Markov chain on Sn ,
defined by a shelf of n books, which are randomly picked by users and then put back
at the left of the shelf after use. In the uniform case (when all books are picked with
the same probability), the determination of the stationary distribution amounts to the
diagonalization of the linear operator on CSn
tn (f ) = f τn
(116)
where
τn = 12 . . . n + 2134 . . . n + 3124 . . . n + · · · + n12 . . . n − 1 ∈ CSn .
(117)
This problem is also an ingredient of the proof of Hivert’s conjecture by Garsia and
Wallach [16]. It can be solved in many different ways. The following one is quite
natural in the context of Noncommutative Symmetric Functions.
We start with the observation that τn is in the descent algebra Σn . Indeed, τn =
D⊆{1} (the sum of permutations having at most a descent at the first position), so
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301
that its representation as a noncommutative symmetric function is S 1,n−1 , a rather
well-understood element.
From this remark, we obtain immediately the eigenvalues of tn . Indeed, according
to Proposition 3.12 of [19], these are the scalar products hn−1 h1 , pλ of ordinary
symmetric functions. Clearly, the scalar product evaluates to m1 (λ), so that the spectrum is 0, 1, 2, . . . , n − 2, n.
Let us now construct the spectral projectors. To this aim, we shall need to evaluate
some polynomials in tn . Let us set Tn = S 1,n−1 and consider the generating function
T=
Tn = S 1 σ 1 .
(118)
n≥0
Since the internal products Tn ∗ Tm are (by definition) 0 for m = n, we have
Tn∗r = T ∗r
(119)
n
and using iteratively the splitting formula ([19], Proposition 2.1)
T ∗ T ∗(r−1) = (S1 σ1 ) ∗ T ∗(r−1) = μ (S1 ⊗ σ1 ) ∗2 Δ T ∗(r−1) ,
(120)
we get the expression
T ∗r = Br (S1 )σ1 ,
(121)
where Br (x) are the Bell polynomials (this is the obvious noncommutative analogue
of the classical formula for the Kronecker powers of the representation of Sn by
permutation matrices).
Using the fact that the coefficients of Bn are the Stirling numbers of the second
kind S(n, k), we obtain in Sym
T ∗ (T − 1) ∗ (T − 2) ∗ · · · ∗ (T − k + 1) = S1k σ1 ,
(122)
and in particular, in degree n,
Tn ∗ (Tn − 1) ∗ (Tn − 2) ∗ · · · ∗ (Tn − n + 1) = S1n
(123)
Tn ∗ (Tn − 1) ∗ (Tn − 2) ∗ · · · ∗ (Tn − n + 2) = S1n ,
(124)
and as well
and since it is plain that
polynomial of Tn is
S1n
∗ Tn =
nS1n ,
so that
Pn (x) = (x − n)
n−2
S1n
∗ (Tn − n) = 0, the minimum
(x − k).
(125)
k=0
This shows that Tn is semisimple, and allows an easy construction of the spectral
projectors.
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J Algebr Comb (2011) 33: 277–312
Let us start with the kernel. The projector is given by f → f ∗ Dn where
Dn =
(Tn − 1) ∗ (Tn − 2) ∗ · · · ∗ (Tn − n + 2) ∗ (Tn − n)
(−1)(−2) · · · (2 − n)(−n)
(126)
but since Tn − n + 1 is invertible, one can take as well
Dn =
(−1)n
(Tn − 1)∗n
n!
(127)
where (x)n = x(x − 1) · · · (x − n + 1), and the star means evaluation with the internal
product. This is a better choice, since we have now a simple generating series for all
these projectors,
(128)
Dn x n = e−xS1 · σx = σx (1 − E)A .
n≥0
Indeed, we have (x − 1)n = (x)n − n(x − 1)n−1 , so that
(−S1 )k
(−1)n
(−S1 )n
(−1)n−1
(T − 1)∗n =
σ1 −
(T − 1)∗ n−1 =
σ1 .
n!
n!
(n − 1)!
k!
n
(129)
k=0
The same reasoning shows that the projectors Dn,k on the eigenspaces of k are
given by the generating series
D(t, u) =
n
t n−k uk Dn,k = e(u−t)S1 σ1 ,
(130)
n≥0 k=0
which is σ1 (tA − (t − u)EA), so that these elements coincide with those defined
by (78).
4.3 Characters of the associated modules
The Frobenius characteristic of the left ideal of CSn generated by the idempotents
δn,k corresponding to Dn,k via the identification σ ↔ Fσ can now be calculated as
follows (compare [32, Corollary 4.2]).
Since δn,k is an idempotent, its characteristic ch(δn,k ) coincides with its cycle index Z(δn,k ). By the Gessel–Reutenauer formula [17], the coefficient of pμ in Z(δn,k )
is equal to Dn,k , Lμ , where F means the commutative image of the noncommutative symmetric function F , and for ν = 1n1 2n2 . . . ,
Lν = hn1 [ 1 ]hn2 [ 2 ] · · · ,
n
=
1
d
μ(d)pn/d
,
n
(131)
d|n
μ denoting here the Moebius function. Hence, the generating function of all the cycle
indexes is
ZY (t, u) = D(t, u; X), L(X, Y ) X = D(t, u; X), L(Y, X) X
(132)
J Algebr Comb (2011) 33: 277–312
303
by the symmetry formula [34]
L(X, Y ) = L(Y, X) =
Lμ (X)pμ (Y ) =
μ
σpn (X)
n (Y )
.
(133)
n≥1
Plugging this last expression into the scalar product and dualizing, we obtain
σpn (t+(u−t)E)
n (Y )
=
n≥1
σ1 ((u − t)Y )
.
1 − tp1 (Y )
(134)
In particular, specializing at Y = E gives that the dimension of CSn δn,k is dn,k , the
number of permutations in Sn with exactly k fixed points.
It is also easy to obtain the expansion of ZY (t, u) as a combination of the
Lμ (Y ). Indeed, writing D(t, u; x) = σ1 ((u − t)EX + tX), we have D(t, u), pμ =
um1 i≥2 t mi , where mi is the multiplicity of the part i in μ. Hence,
Lμ .
(135)
Z(δn,k ) =
m1 (μ)=k
We note that this is the quasi-symmetric generating function of the permutations with
exactly k fixed points. Note that Z(D(t)) is the commutative image of the generating
series of desarrangements (23).
4.4 q-derangement numbers
From the above considerations, one can easily derive a (known) closed formula for
the q-derangement numbers (compare [32, Theorem 4.5])
dn (q) :=
q maj σ .
(136)
σ ∈Dn
Indeed,
dn (q) =
Fσ ,
σ ∈Dn
=
q
τ ∈Sn
q
maj(I )
maj(τ )
Gτ
FQSym
FC(σ ) , RI =
σ ∈Dn |I |=n
Lμ , Kn (q) .
(137)
m1 (μ)=0
Hence,
n≥0
dn (q)
X
xn
= σ1
x n ln , σ 1
(q)n
1−q
n≥2
λ−x
X
1
x
=
, σ1
= λ−x
1−
1 − xp1
1−q
1−q
1−q
1−q
1 − xq n
=
1−x −q
n≥0
−1
(138)
304
J Algebr Comb (2011) 33: 277–312
so that finally [36]
dn (q) = [n]!
n
(−1)k
[k]!
k=0
k
q (2 ) .
(139)
4.5 Characters from Lie idempotents
The expression
ch(δn,k ) =
(140)
Lλ ,
m1 (λ)=k
is also a consequence of the following type of expressions:
Dn,k =
Eλ [π]
(141)
m1 (λ)=k
in the notation of [19, Theorem 3.16], for some sequence π = (πn ) of Lie idempotents in descent algebras. Indeed, [19, Theorem 3.21], implies then that the character is given by (140). We have already seen one such expression with πn = ζn , the
Zassenhaus idempotents. We can also write one involving the Hausdorff series. Writing as usual
Φ=
φn = log σ1
(142)
n≥1
(the Solomon idempotents), we have
Dn = e−φ1 eΦ = eH (−φ1 ,Φ)
(143)
n≥0
where H is the Hausdorff series. Taking π1 = φ1 and πn = Hn (−φ1 , Φ) for n ≥ 2,
we obtain a sequence of Lie idempotents (see, e.g., [19], Theorem 3.1), from which
it is easy to build a decomposition of the identity
(144)
πn ,
σ1 = eπ1 exp
n≥2
and more explicitly,
Sn =
1
r!s!
r+s=n
s ,J
π1
.
(145)
(J )=r,|J |=n−s,1∈J
This gives in particular the decomposition (141), with, for a partition λ such that
m1 (λ) = 0,
1 I
Eλ (π) =
π ,
(146)
(λ)!
I ↓λ
where I ↓ λ means that the nondecreasing rearrangement of the composition I is the
partition λ.
J Algebr Comb (2011) 33: 277–312
305
4.6 Eigenbases of tn
From Proposition 7.4 of [11], we know the image a projector of the type (146). It is
formed of weighted symmetrizations of Lie elements. With the above πn , the distribution is uniform, so that the kernel consists in ordinary symmetrized products of Lie
elements. Concretely, a basis of Ker tn in CSn is, for example,
(147)
(γ1 θλ1 , γ2 θλ2 , . . . , γr θλr )
where (a, b, c) = abc + acb + bac + bca + cab + cba, and so on (symmetrized
products), the γk are the minimal representatives of the cycles of a derangement,
θn = [[· · · [1, 2], 3], · · · n] is a Dynkin element, and λ runs over partitions without
part 1.
For example, a basis of Ker t4 of dimension d4 = 9 is given by the elements
[[[1, a], b], c] with abc running over permutations of 234, for L4 , and by the three
symmetrized products ([1, 2], [3, 4]), ([1, 3], [2, 4]), and ([1, 4], [2, 3]) for L22 .
Bases of the other eigenspaces are obtained by the same process, using weighted
symmetrizations as indicated in [11]. Indeed, (145) shows that a basis of the
eigenspace with eigenvalue s is given by
(γ1 θλ1 , γ2 θλ2 , . . . , γr θλr ) · (j1
j2
···
js )
(148)
where γk are the minimal representatives of the cycles of length at least 2 a permutation of cycle type (λ, 1s ) having s fixed points j1 , j2 , . . . , js .
To continue with n = 4, a basis of the 1 eigenspace is [[i, j ], k] · l (i < j, k, ij kl a
permutation of 1234), dimension 8, and a basis of the 2-eigenspace is given by
[i, j ] · (kl + lk),
i < j, k < l
where ij kl is a permutation of 1234 (dimension 6). Finally, the 4-eigenspace is one
dimensional and generated by the full symmetrizer.
Using the ζn instead of the πn , we can replace symmetrized products by ordinary products of homogeneous Lie polynomials taken in nondecreasing order of the
degrees.
The idempotents δn,k have been first studied by M. Schocker [32] (apparently unaware of previous works on the subject and of their relation with the Tsetlin library).
4.7 A basis of FQSym
We have seen that the (1 − E)-transform is a bialgebra morphism in FQSym. Hence,
its image FQSym is a Hopf subalgebra. The Sn -module Δn,k generated by δn,k can
be identified with
FQSymn = FQSymn ∗ Dn,k ,
(149)
dim FQSymn = dn .
(150)
so that
306
J Algebr Comb (2011) 33: 277–312
It is therefore desirable to find a basis of FQSym labeled by derangements, or some
other set of permutations naturally in bijection with these. As we shall see, the natural transformation involved here is simply a version of Foata’s first fundamental
transformation [14].
Let γn be the cycle
γn := n 1 2 . . . n − 1,
(151)
so that
Sγn = Tn = S1 Sn−1 = Rn + R1,n−1 =
Fσ .
(152)
σ ∈1 23...n
Since the -transform is a morphism for the product of FQSym,
Sγn ∗ σ1 = S1 Sn−1 = 0,
(153)
and, for any permutation σ ∈ Sn ,
(Fσ ∗ Sγn ) = Fσ ∗ Sγn ∗ σ = 0.
(154)
Recall that i is a left-right minimum of σ if
σj > σi
for all j < i.
(155)
Let Xn be the set of permutations of Sn such that σ · 0 does not have two consecutive
left-right minima (that is, σ does not end by 1 and does not have two consecutive
LR-minima), and let Yn = Sn \Xn .
Lemma 4.1 For σ ∈ Yn , write
σ · 0 = · · · σi σi+1 · · ·
(156)
where i and i + 1 is the first pair of consecutive LR-minima, and let
σ = σi · σ1 · · · σˆi · · · σn
(157)
be the permutation obtained by moving σi at the first position, leaving the remaining
letters unchanged, and removing the zero in the end. Then
Fσ ∗ Sγn = Fσ +
Fτ ,
(158)
τ ∈T
where the permutations of T are lexicographically smaller than σ .
Proof The expression
Fσ ∗ Sγn =
τ ∈σi
Fτ
(159)
σ1 ...σˆi ...σn
contains Fσ and, since σi is a LR-minimum, σ is the maximal element of the previous
sum.
J Algebr Comb (2011) 33: 277–312
307
For a permutation σ with LR-minima i1 , . . . , ip , let
φ(σ ) = (σ1 . . . σi1 −1 )(σi1 . . . σi2 −1 ) . . . (σip . . . σn ),
(160)
where each parenthesis represents a cycle. For example, with σ = 62781453,
φ(σ ) = (6)(278)(1453) = 47153682.
(161)
This is Foata’s first fundamental transformation (up to reversing the order on the
integers), hence a bijection. Clearly, φ(σ ) has fixed points whenever σ ∈ Yn , so φ
induces a bijection between Xn and derangements of Sn .
From Lemma 4.1, we see that the elements
Fσ
(162)
σ ∈Xn
span FQSymn . Since |Xn | = dn = dim FQSymn , we have finally
Theorem 4.2 The Fσ for σ ∈ Xn form a basis of FQSym .
The sets Xn have an interesting structure.
Theorem 4.3 The set Xn is an ideal of the left weak order on Sn . Its maximal elements are the left-shifted concatenations
wI := wi1 · · · wir ,
(163)
where wi := 1 i i − 1, . . . , 2, composition I has no part 1, and α β = α[ ] · β if
β ∈S .
Proof To show that Xn is an ideal, we will prove that if si denotes the elementary
transposition (i, i + 1), then σ ∈ Yn and inv(si σ ) = inv(σ ) + 1, implies si σ ∈ Yn .
If σk = r > s = σk+1 are consecutive LR-minima of σ , they will remain so for
si σ , unless i = r − 1, r, s − 1, or s. Since si σ has one inversion more than σ , we can
exclude the case i = r − 1: r being a LR-minimum, r − 1 cannot be to the left of r
in σ . We can also exclude i = s − 1 for the same reason. If i = r, then r is exchanged
with r + 1, which has to be to its right in σ , so that again σk and σk+1 are consecutive
LR-minima in si σ . The same reasoning applies with i = s. Hence, Yn is a coideal,
and consequently Xn is an ideal.
Now, the elements wI are clearly in Xn when I has no part 1, and any exchange
of consecutive values creating an inversion in such a wI would create a pair of consecutive LR-minima. So these wI are maximal elements of the ideal Xn .
Conversely, consider σ ∈ Xn maximal. Then consider the suffix s of σ beginning
with 1. The maximality condition of σ implies that if t belongs to that suffix, then
t − 1 also belongs to it. So this prefix is a permutation of an S|s| , then should be
1|s| . . . 2. The same now works by induction on the permutation τ defined by σ =
τ s.
308
J Algebr Comb (2011) 33: 277–312
For example, with n = 5, we get the following three maximal elements of Xn :
15432, 35412, 45132.
(164)
The same proof can be adapted to the case of permutations that are images by φ of
(k)
permutations with at most k fixed points. Let Xn be the image by φ of permutations
(k)
with at most k fixed points. Then Xn is the set of permutations with at most k
consecutive LR-minima.
(k)
Theorem 4.4 The set Xn is a ideal of the left weak order on Sn . Its maximal elements are the wI where I runs over compositions with
• either k − 1 ones and the remaining parts equal to 2,
• or exactly k ones.
(k)
Proof The fact that Xn is an ideal comes from the same idea as before: all permutations greater than a given permutation σ for the left weak order have LR-minima at
the same position.
By the same argument as in the previous theorem, the maximal elements must be
some wI , where I has at most k ones. Now, it is clear that
(165)
wI < wJ ,
in the left weak order iff I can be obtained from J by gluing parts equal to 1 with
their next part. So the compositions described in the statement are definitely maximal
elements. And since all compositions with at most k ones can be obtained from these
ones by the gluing process, this ends the proof.
(k)
Here is a table (Table 2) of the number of maximal elements of Xn . Note that
the first column is obviously given by Fibonacci numbers since these indeed count
the number of compositions of n in parts at least 2. The other columns are not known
to [35] and neither is the sequence of row sums.
Table 2
n\k
0
1
2
3
4
5
6
7
1
0
1
2
1
1
1
3
1
2
2
1
4
2
3
3
3
5
3
5
6
4
4
1
6
5
9
9
10
5
5
1
7
8
15
19
14
15
6
6
1
8
13
27
31
34
20
21
7
7
8
1
1
J Algebr Comb (2011) 33: 277–312
309
But there exists a simple formula giving the number of maximal elements, coming
directly from their characterization:
n−k
2 (n + k − 1)/2
+k n−k− −1
mn,k :=
+
,
k−1
k
−1
(166)
=0
with the convention that a binomial coefficient with entries not in the natural numbers
is zero.
4.8 Other bases of FQSym
Conjecture 4.5 Let < be the order on permutations defined by
σ< τ
⇐⇒
φ(σ ) <lex φ(τ ).
(167)
Then the matrix of Sσ of the S basis is triangular. Moreover, the diagonal values are
1 for the elements of Xn and 0 for Yn .
For example, here are the matrices for n = 2, 3, and 4 (Fig. 1) where the zero
entries have been represented by dots to enhance readability.
The permutations are ordered as follows:
[21, 12],
[321, 312, 231, 123, 132, 213].
(168)
[4321, 4312, 4231, 4123, 4132, 4213, 3421, 3412,
2341, 1234, 1243, 2314, 2431, 1423, 3241, 2134,
3142, 1324, 1432,
⎛
.
⎜.
⎜
⎜.
. −1/2
⎜
⎜.
.
1
⎜
⎝.
.
2413, 2143, 3214, 1342, 3124].
⎞
. . 1/3 −1/3 2/3
. . −1
.
−1 ⎟
⎟
. .
.
.
. ⎟
⎟.
. . 1
.
1 ⎟
⎟
. .
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1
−1 ⎠
. .
.
.
.
(169)
(170)
5 Other combinatorial Hopf algebras
5.1 The algebras PQSym and CQSym
There is an internal product on PQSym extending that of WQSym∗ [26]. The transform is defined in PQSym (it contains Sym as a subalgebra), but PQSym ∗
WQSym∗ ⊆ WQSym∗ , so that PQSym = W , and we get nothing new.
Similarly, for the Catalan algebra CQSym [28], we have
CQSym ∗ Sym ⊆ Sym,
(171)
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Fig. 1 The matrix of in the S basis of FQSym
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⎜.
⎜.
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⎜.
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⎜.
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⎟
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⎟
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⎟
⎟
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⎟
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310
J Algebr Comb (2011) 33: 277–312
J Algebr Comb (2011) 33: 277–312
311
so that
CQSym = Sym .
(172)
5.2 The algebra of planar binary trees PBT
The Loday–Ronco algebra of planar binary trees is not stable by the -transform.
Since PBT is the subalgebra of FQSym generated by the S σ where σ avoids the
pattern 132 (see [13]), we have, for example,
2
S 213 = S 123 − S 132 − S 312 + S 321 ∈
/ PBT.
3
(173)
However, PBT is a well-defined Hopf subalgebra of FQSym.
Conjecture 5.1 The algebra PBT is free over the set PT , where T runs over trees
with at least two nodes, and such that the right subtree of the root is empty.
In particular, the conjecture implies that the dimension of the homogeneous com
ponents PBTn are given by the Fine numbers [8, 35], sequence A000957:
1, 0 1 2 6 18 57 186, 622, . . . .
(174)
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